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Multigraphs and Time Ordered Isserlis-Wick formulae

Sergio Cacciatori, Batu Güneysu, Sebastian Wündsch

TL;DR

The paper addresses the problem of evaluating time-ordered integral expressions $\int_{0\le s_1\le\cdots\le s_n\le1}\mathbb{E}[\prod_{k=1}^n Q(X(s_k))]\,ds_1\cdots ds_n$ for an $m$-variate zero-mean Gaussian process with covariance $\mathrm{Cov}(X(s),X(t))=f(s,t)I_m$. It develops two complementary formulations: a purely algebraic generalization of Wick's theorem using multisets and a graph-based, multigraph labeling (Feynman-diagrammatic) approach. The main contributions are explicit combinatorial constructs $C(M,A)$ and $C(\Gamma)$ that yield two equivalent expansions of the target integral, one over upper-triangular matrices and one over collections of multigraphs with prescribed vertex degrees. These results link Isserlis-Wick-type expansions to Feynman-Kac-type representations, offering tractable analytic tools for heat-kernel and exponential-functionals in Gaussian settings.

Abstract

Given a m-dimensional Gaussian process and polynomial m variables with real coefficients, we calculate the induced path odered exponenial in two different ways: one is purely algebraic in spirit and the other one is diagrammatic in spirit and uses multigraph labelings (and is inspired by the use of Feynman diagrams in quantum field theory).

Multigraphs and Time Ordered Isserlis-Wick formulae

TL;DR

The paper addresses the problem of evaluating time-ordered integral expressions for an -variate zero-mean Gaussian process with covariance . It develops two complementary formulations: a purely algebraic generalization of Wick's theorem using multisets and a graph-based, multigraph labeling (Feynman-diagrammatic) approach. The main contributions are explicit combinatorial constructs and that yield two equivalent expansions of the target integral, one over upper-triangular matrices and one over collections of multigraphs with prescribed vertex degrees. These results link Isserlis-Wick-type expansions to Feynman-Kac-type representations, offering tractable analytic tools for heat-kernel and exponential-functionals in Gaussian settings.

Abstract

Given a m-dimensional Gaussian process and polynomial m variables with real coefficients, we calculate the induced path odered exponenial in two different ways: one is purely algebraic in spirit and the other one is diagrammatic in spirit and uses multigraph labelings (and is inspired by the use of Feynman diagrams in quantum field theory).
Paper Structure (2 sections, 7 theorems, 96 equations)

This paper contains 2 sections, 7 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Main Results

Key Result

Theorem 1.1

If $Z=(Z_1,\dots,Z_m)\in\mathbb{R}^m$ is a zero mean Gaussian random vector, then one has if $m$ is odd, and if $m$ is even.

Theorems & Definitions (13)

  • Theorem 1.1
  • Example 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof : Proof of Theorem \ref{['prop']}
  • Corollary 2.4
  • Theorem 2.5
  • ...and 3 more